Pump action seems to be a faster way to cycle a weapon compared to bolt action and it's a rather simple action, unlike straight-pull bolt which are more prone to malfunction. It almost seems like a direct upgrade to me.
Since we have magazine-fed pump action shotguns nowadays, It's probably possible to do the same with rifles. Even if we only had tubular magazines, the Lebel rifle actually had a tubular magazine with spitzer (pointy) bullets that was designed to prevent cartridges from firing the one in front of it in the magazine.
I know that nowadays this would be a novelty because of the availability of semi-auto rifles but why couldn't it replace bolt-action rifles?